Experimental Observation of Vortex Rings in a Bulk Magnet
Claire Donnelly, Konstantin L. Metlov, Valerio Scagnoli, Manuel Guizar-Sicairos, Mirko Holler, Nicholas S. Bingham, Jörg Raabe, Laura J. Heyderman, Nigel Cooper, Sebastian Gliga
EExperimental Observation of Vortex Rings in a Bulk Mag-net
Claire Donnelly , , , Konstantin L. Metlov , , Valerio Scagnoli , , Manuel Guizar-Sicairos , MirkoHoller , Nicholas S. Bingham , , J¨org Raabe , Laura J. Heyderman , , Nigel Cooper and Sebas-tian Gliga Cavendish Laboratory, University of Cambridge, JJ Thomson Ave, Cambridge CB3 0HE, UK. Laboratory for Mesoscopic Systems, Department of Materials, ETH Z ¨urich, 8093 Z ¨urich,Switzerland. Paul Scherrer Institute, 5232 Villigen PSI, Switzerland. Donetsk Institute for Physics and Engineering, R. Luxembourg 72, Donetsk 83114, Ukraine. Institute for Numerical Mathematics RAS, 8 Gubkina str., 119991 Moscow GSP-1, Russia.
Vortex rings are remarkably stable structures occurring in numerous systems: for examplein turbulent gases, where they are at the origin of weather phenomena ; in fluids with im-plications for biology ; in electromagnetic discharges ; and in plasmas . While vortex ringshave also been predicted to exist in ferromagnets , they have not yet been observed. UsingX-ray magnetic nanotomography , we imaged three-dimensional structures forming closedloops in a bulk micromagnet, each composed of a vortex-antivortex pair. Based on themagnetic vorticity, a quantity analogous to hydrodynamic vorticity, we identify these con-figurations as magnetic vortex rings. While such structures have been predicted to exist astransient states in exchange ferromagnets , the vortex rings we observe exist as stable, static a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p onfigurations, whose stability we attribute to the dipolar interaction. In addition, we ob-serve stable vortex loops intersected by magnetic singularities , at which the magnetisationwithin the vortex and antivortex cores reverses. We gain insight into the stability of thesestates through field and thermal equilibration protocols. These measurements pave the wayfor the observation of complex three-dimensional solitons in bulk magnets, as well as for thedevelopment of applications based on three-dimensional magnetic structures. In magnetic thin films, vortices are naturally occurring flux closure states, in which the mag-netisation curls around a stable core, where the magnetisation tilts out of the film plane
8, 9 . Thesestructures have been studied extensively over the past decades due to their intrinsic stability andtheir topology-driven dynamics , which are of both fundamental and technological interest.Antivortices, the topological counterpart of vortices, distinguish themselves from vortices by anopposite rotation of the in-plane magnetization that is quantified by the index of the vector field –which is equal to the winding number of a path traced by the magnetisation vector while movingin the counterclockwise direction around the core . While vortices have a circular symmetryof the magnetisation (figure 1a), antivortices only display inversion symmetry about the center (figure 1b), resembling saddle points in the vector field. Experimental studies of magnetic vor-tices and antivortices have mostly been restricted to two dimensional, planar systems, in whichvortex-antivortex pairs have a natural tendency to annihilate , unless they are part of larger, stablestructures, such as cross-tie walls .In bulk ferromagnets, the existence of transient vortex rings, that take the form of localised2olitons and are analogous to smoke rings, has been predicted , but such structures have so farnot been observed. Just as vortex rings in fluids are characterised by their vorticity, ferromagneticvortex ring structures can be identified by considering the magnetic vorticity. By analogy withfluid vorticity, the magnetic vorticity is a vector field, which can be defined as : Ω α = 18 π (cid:15) αβγ (cid:15) ijk n i ∂ β n j ∂ γ n k (1)where n α ( r , t ) is a component of the unit vector representing the local orientation of the magneti-sation, α indicates the vorticity component, and (cid:15) αβγ is the Levi-Civita tensor, summed over threecomponents x, y, z . The magnetic vorticity vector Ω represents the topological charge flux (orSkyrmion number ) density. Integrating the magnetic vorticity over a closed two-dimensional sur-face, results in a scalar value (cid:82) Ω · d S = N corresponding to the skyrmion number, which gives thedegree of mapping of the magnetization distribution to an order parameter space described by thesurface of an S sphere. When N = 1 , the target sphere is wrapped exactly once and each directionof the magnetisation vector is present. The magnetic vorticity vector Ω is therefore non-vanishingin the vicinity of the cores of vortices or antivortices, and is represented in Figure 1a-d for vorticesand antivortices with different polarisations (the polarisation is the orientation of the magnetisa-tion within the core). The vorticity vector is aligned parallel to the polarisation of a vortex (a,c)and antiparallel to the polarisation of an antivortex (b,d), indicating that it is dependent upon thedirection of the magnetisation in the core as well as the index of the structure. Consequently, avortex-antivortex pair with parallel polarisations, exhibit opposite vorticities, that circulate in aclosed loop (Figure 1e).Here, we use the magnetic vorticity to locate and identify magnetic structures within a three-3imensional magnetic micropillar, that are imaged using hard X-ray magnetic nanotomography.Within the bulk of the pillar, we find two types of vorticity loops. The first is characterised by a cir-culating magnetic vorticity forming vortex rings, analogous to smoke rings. These magnetic vortexrings consist of vortex-antivortex pairs with parallel polarisations, as in Figure 1e. Consequently,the magnetisation distribution does not wrap the order parameter space and the pair belongs tothe same topological sector as a uniformly magnetised domain. The second type of loop containssingularities, or Bloch points , at which the vorticity abruptly reverses its sign, thus modifying thetopology of the vortex-antivortex pairs. Calculating preimages of the observed structures indicatesthat the vortex rings display concentric pre-images corresponding to a trivial knot, with a vanishingHopf index, while structures containing Bloch points have preimages similar to recently observed‘toron’ structures in anisotropic fluids .The hard X-ray magnetic nanotomography setup is illustrated in Figure 1f. During the mea-surement, high resolution X-ray projections of the sample were measured with dichroic ptychography for 1024 orientations of the sample with respect to the X-ray beam. The photon energy of thecircularly-polarised X-rays was tuned to the Gd L edge and, by exploiting the X-ray magneticcircular dichroism effect, sensitivity to the component of the magnetisation parallel to the X-raybeam was obtained. In order to gain access to all three components of the magnetisation, X-rayprojections were measured for different sample orientations about the tomographic rotation axisfor two different sample tilts. The internal magnetic structure was obtained using an iterative re-construction algorithm , which has been demonstrated to offer a robust reconstruction of nanoscalemagnetic textures . Further experimental details are given in the Methods section.4sing this method, we image the magnetic structure of a bulk GdCo ferrimagnetic cylinderof diameter 5 µ m , in which the coupling between the two antiparallel magnetic sublattices leadsto an effective soft ferromagnetic behavior . The lowest energy state of such a magnetic cylinderis expected to consist of a single vortex . In our system, the size of the pillar is large enough toreduce the role of surface anisotropy, supporting the stabilisation of more complex, metastablestates, that can include a large number of vortices, anti-vortices, domain walls and singularities .We compute the magnetic vorticity Ω from the reconstructed magnetisation following equa-tion (1). Regions of large vorticity are plotted in Figure 1g, where a number of ‘tubes’ and loopscorresponding to the cores of vortices and antivortices are visible. In addition, unlike in incom-pressible fluids, where the divergence must vanish, a non-zero divergence of the magnetisation, m , is allowed in ferromagnets, given that Maxwell’s equations only exclude the divergence of B .In this way, computing the magnetic vorticity also allows us to locate singularities of the mag-netisation – known as Bloch points – within the system, which are characterised by a very largedivergence of the magnetic vorticity, ∇ · Ω , due to the local variation in the orientation of themagnetisation. Here, Bloch point and anti-Bloch points are identified by positive (red) and neg-ative (blue) ∇ · Ω , as plotted in Figure 1h. Within the pillar, we find an equal number of Blochpoints and anti-Bloch points, indicating that the singularities originated in the bulk of the structure,where they can only be created in pairs. As a result, it appears that sample boundaries, throughwhich a single Bloch point could be injected, most likely did not play a role in the formation of theobserved structures. 5igure 1: Measuring and reconstructing the internal magnetic structure and the magnetic vortic-ity within a GdCo pillar. a-d) Schematic representation of the magnetic vorticity Ω , shown inpurple and orange arrows, for a number of vortex and antivortex configurations with different po-larisations (red, blue). The vorticity of a vortex-antivortex pair with same polarisation is shown in(e). f) Schematic representation of the experimental setup: tomographic projections with magneticcontrast are measured using dichroic ptychography for the sample at several different orientationswith respect to the X-ray beam. Measurements were performed with the sample at two different tiltangles: 30 ◦ (transparent green) and 0 ◦ (blue). g) Plotting regions of significant magnetic vorticity,we locate a network of structures, and h) plotting regions of high divergence of the vorticity ∇ · Ω ,we locate Bloch points (red) and anti-Bloch points (blue), which respectively have positive andnegative divergence. 6igure 2: Structure of a vortex ring with circulating magnetic vorticity. a) A vorticity ‘loop’ isidentified next to a vortex by plotting an isosurface corresponding to m x = ± . The in-planemagnetisation within a two-dimensional slice through the loop is plotted using streamlines, reveal-ing two vortices enclosing an antivortex, with the loop consisting of a vortex-antivortex pair. Thecolourmap indicates the value of m x , which corresponds to the direction of the magnetisation inthe core (polarisation), showing that the vortex and the antivortex within the loop have the samepolarisation. b) Mapping the vorticity (represented both by the arrows and the colourmap), revealsthat the loop exhibits a circulating vorticity and is a vortex ring. The vorticity map equally indicatesthat, in the nearby extended vortex, the vorticity abruptly reverses, corresponding to the presenceof a Bloch point. Note that the plotted structures have a relatively low vorticity, with | Ω | = 0 . (with the exception of the Bloch point). c) Plotting preimages for different directions reveals anumber of closed loops, that, when the vorticity is plotted, are seen to correspond to vortex rings(insets). d) In the vicinity of the vortex loop in a), preimages for neighbouring directions are notlinked, indicating a Hopf index of zero. 7mong the plotted structures in Figure 2, there appear a large number of three-dimensional‘loops’, that resemble the vortex-antivortex pair schematically illustrated in Figure 1e. We firstconsider the case of one such loop that is identified using the m = + ˆx pre-image in Figure 2a,where m = | M | /M s is the reduced magnetisation, and M s is the saturation magnetisation. Thisloop is located in the vicinity of a single vortex extending throughout the pillar and whose polari-sation equally points along the + ˆx direction in the shown slice. Considering the magnetisation inthe y − z plane, represented by streamlines in Figure 2a, we identify a bound state consisting oftwo vortices separated by an antivortex, analogous to a cross-tie wall. The loop itself is embeddedwithin a quasi-uniformly magnetised region ( m = + ˆx , red) and therefore the vortex and antivor-tex have same polarisations, as shown schematically in Figure 1e. When the magnetic vorticityvector Ω is plotted, see Figure 2b, it exhibits a unidirectional circulation around the loop, directlycomparable to the schematic in Figure 1e. This structure is similar to a vortex ring in a fluid, whichalso corresponds to a loop in the hydrodynamic vorticity. Such vorticity loops have been predictedto exist as propagating solitons in exchange ferromagnets . In contrast, the vortex loops observedhere are static and stable at room temperature over the duration of our measurements. We notethat the diameter of the vortex ring, i.e. the average distance between the vortex and antivortexcores in the y − z plane, is approximately
370 nm , and is comparable to the diameter of most othervortex rings present inside the pillar. Interestingly, this loop (along with a number of similar vortexrings in the sample) occurs in the vicinity of a singularity: indeed, the neighbouring vortex in thecross-tie structure contains a Bloch point, which can be located in Figure 2b where the vorticity,(and the magnetisation in the vortex core) abruptly reverses direction. There is a priori no topo-8ogical requirement for the presence of a Bloch point in proximity of the vortex loop and despitethe observed correlations, our static observations do not allow for the determination of a causalrelationship between the presence of both structures.We gain further insight into the topology of these vortex loops by plotting preimages cor-responding to a number of directions of the magnetisation in the vicinity of the vortex-antivortexpair. The preimage corresponding to the + ˆx direction, i.e. m x = +1 , is plotted in light greenin Figure 2d, along with additional preimages corresponding to directions indicated in the insetthat form an ensemble of closed-loop preimages. The plotted loops do not link, indicating thatthe vortex ring has a Hopf number H = 0 . Indeed, the vicinity of the H = 0 structure containsonly preimages representing directions close to the + ˆx direction and, consequently, do not coverthe S sphere, meaning that the magnetisation can be smoothly unwind into a single point on thesphere . Hence, these vortex rings belong to a class of non-topological solitons . In the Methods(figure M1c), we have developed an analytic model of such a soliton, qualitatively reproducing theobserved features, vorticity and pre-images.In addition to vortex rings, we also identify vorticity loops containing sources and sinks ofthe magnetisation, due to the presence of Bloch points. The magnetic structure of one such loop isshown in Figure 3a, where the colourscale indicates the polarisation ( ± ˆx ) and the magnetisationin a plane of the loop is represented by streamlines, revealing a vortex-antivortex pair. At twopoints within the loop, the polarisation along the vortex and antivortex cores reverses with thecolour changing from blue to red. Consequently, the vorticity does not circulate around the loop,9igure 3: Structure of a vorticity loop containing magnetization singularities. a) The vorticityloop is identified by its relatively high magnetic vorticity. The magnetic configuration in a two-dimensional slice through the loop is plotted using streamlines to represent the in-plane magneti-sation, with the colour indicating the out-of-plane magnetisation component ± m x and revealingthat the loop consists of a vortex and antivortex pair. Within the loop, the x direction of the mag-netisation, i.e. the core polarisation, switches from positive (red) to negative (blue) at two points,indicated by the orange and green boxes. b) Plotting the magnetic vorticity reveals that this is infact not a closed loop, but an “onion” state, with the vorticity direction reversing at the same twopoints. These locations correspond to singularities of the magnetisation (c,d) and, consequently, ofthe magnetic vorticity (e,f). g), the preimages corresponding to the Cartesian axes ± ˆx (light/darkgreen), ± ˆy (light/dark red), and ± ˆz (light/dark blue) are plotted, which reveal an onion-like state,with all preimages meeting at the singularities. 10ut instead assumes an asymmetric onion-like structure, with the vorticity flowing out from asource (green box in Figure 3b) and into a sink (orange box in Figure 3b). The structure of themagnetisation in the vicinity of the singularities is plotted in Figures 3c,d. In the vicinity of thevorticity sink (Figure 3e), the magnetisation structure (shown in Figure 3c) corresponds to thatof a contra-circulating Bloch point (or anti-Bloch point) with skyrmion number − . Aroundthe vorticity source (Figure 3f), the magnetisation structure (Figure 3d) corresponds to that of acirculating Bloch point with skyrmion number +1 . Two features of this loop are particularlynoteworthy. First, the singularities are not linked to the generation and annihilation of a vortexand antivortex with opposite polarisations, as has been reported for dynamic processes . Instead,the pair consists of two half-vortex rings connected by the Bloch points, which locally leads to areversal of the vorticity along the vortex and the antivortex cores, as seen in Extended Data, FigureM2. Second, while singularities often mediate dynamic processes and have been predicted duringmagnetisation dynamics
28, 29 as well as during magnetic field reconnection in plasma physics , theobserved structures are inherently static. In Ref. 6, Bloch points were observed at the locationswhere a vortex core intersected a domain wall. Similarly, we find that the Bloch point pair islocated at the intersection of the vortex-antivortex loop with a domain wall separating regions ofopposite m x .We gain further insight into the topology of the vortex-antivortex loop containing singu-larities by plotting preimages corresponding to a defined set of directions, or points, on the S sphere. In particular, we plot regions of the magnetisation aligned along ± ˆx (bright/ dark green), ± ˆy (bright/ dark red), and ± ˆz (bright/ dark blue) in Figure 3g, which can be seen to form a11igure 4: Magnetic vorticity plots measured for the GdCo micropillar at remanence showing theeffect of different field histories on the vortex-antivortex structures a) following the application ofa 7 T saturating field and c) following saturation and field cooling. A small number of vortex loopslike those in figure 2 are present at remanence after the application of a saturating magnetic field,shown in b), however none are observed following the thermal annealing procedure.three-dimensional onion state, with all directions of the magnetisation meeting at the singularitiesschematically indicated by green and orange circles, corresponding to the anti-Bloch point andBloch point, respectively. The preimages resemble those found to correspond to ‘torons’, whichhave recently been observed in chiral liquid crystals and anisotropic fluids . In the methods,we present an analytical model of different micromagnetic configurations with similar pre-images,allowing us to reproduce and, consequently, understand the experimental observations.We explore the stability of the observed vorticity loops by applying two different field andthermal protocols on a similar GdCo micropillar, and performing magnetic X-ray nanotomog-raphy at remanence following each protocol. In the first protocol, we apply a magnetic fieldalong the long axis of the pillar at room temperature, and image the resulting remanent configura-12ion. The applied field is above the measured sample saturation field of ca. ∼ . A plot of themagnetic vorticity (see figure 4a) reveals a large number of vortices and antivortices, as well asmagnetic singularities (shown in Methods and figure M4 at remanence). By plotting pre-imagescorresponding to different directions of the magnetisation, we observe a small number of vortexloops, two of which are shown in figure 4b. The presence of these vortex loops after the applicationof a saturating magnetic field indicates that the loops can nucleate spontaneously, and therefore donot require a specific field protocol to prepare them. Secondly, we heat the sample to
400 K whileapplying a magnetic field. The sample is then field cooled and the field gradually removedafter the sample reached room temperature. This annealing procedure is reminiscent of those usedto expel defects in single-crystals in order to increase their purity. A plot of the vorticity, shown infigure 4c, reveals a noticeably smaller number of structures with non-zero vorticity. Importantly,we do not find any vortex loops, indicating that these are metastable states which are more effi-ciently destroyed through thermal annealing. Quantitatively, the average vorticity value followingfield cooling is half the value following only the application of a 7 T field, and the total number ofBloch points is roughly halved ( vs. Bloch points, as seen in figure M4).Although the vortex rings we observe are topologically trivial structures and have a Hopfindex of zero, they are surprisingly stable. We attribute their stability to interactions with sur-rounding magnetization structures, which ensure that they are, for example, embedded in largercross-tie structures or pinned at the intersection with domain walls, resulting in loops intersectedby Bloch points. Moreover, the magnetostatic interaction clearly plays an important role in thestabilisation of these structures, ensuring that our observations of stable localised solitons do not13ontradict the Hobart-Derrick theorem for an exchange ferromagnet that requires non-linearities(such as intrinsic chirality in the presence of Dzyaloshinskii-Moriya interaction) to set a scale forlocalised magnetisation non-uniformities. We note that chirality has been demonstrated in a similarbulk amorphous system through the inclusion of structural inhomogeneities . We expect that suchsystems could host topologically non-trivial solitons, such as knots with a higher Hopf number, aswell as torons, following predictions for chiral magnetic heterostructures
32, 34, 35 , analogous to thereported observations in chiral liquid crystals and ferrofluids
26, 36 .Finally, very recent advances in time-resolved X-ray magnetic laminography open the pathto investigating the dynamics of three-dimensional magnetic configurations. As well as probingresonant dynamics, it is possible that investigations of the stability and motion of three-dimensionalvortex rings could reveal behaviour analogous to the Kelvin motion of two-dimensional vortex-antivortex pairs . Likewise, we expect that the magnetic vortex loops discovered here con-taining singularities will also display compelling dynamics, with implications for the fundamentalunderstanding of the role of singularities in magnetic processes. The study of the conditions forthe formation of three-dimensional magnetic structures, and of their stability and dynamics, is ex-pected to lead to new possibilities for the controlled manipulation of the magnetisation that couldbe relevant for technological applications requiring complexity, such as neuromorphic computing or new proposals for three-dimensional data storage .14 MethodsSample Fabrication
The samples investigated were both
GdCo micropillars of diameter µ m that were cut from a larger nugget of GdCo using a focused ion beam in combination with amicromanipulator, and mounted on top of OMNY tomography pins . X-ray ptychographic tomography
Hard X-ray magnetic tomography was performed at the cSAXSbeamline at the Swiss Light Source, Paul Scherrer Institut, using the flexible tomographic nanoimaging (flOMNI) instrument . Part of the data presented in this manuscript (the central vortexcontaining the Bloch point in Figure 2a,b) formed part of the dataset presented in Ref. 6. All otherdata is shown and analysed for the first time here.Two dimensional tomographic projections were measured with X-ray ptychography, a coher-ent diffractive imaging technique allowing access to the full complex transmission function of thesample
45, 46 . For X-ray ptychography, an X-ray illumination of approximately µ m was definedon the sample, and ptychography scans were performed by measuring diffraction patterns on aconcentric grid of circles with a radial separation of . µ m for a field of view of × µ m and × µ m for the untilted and tilted sample orientation, respectively. The projections were recon-structed using 500 iterations of the difference map and 200 iterations of the maximum likelihoodrefinement using the cSAXS PtychoShelves package .To probe the magnetisation of the sample, X-rays tuned to the Gd L edge with a photon en-ergy of .
246 keV were chosen to maximise the absorption XMCD signal . Circularly polarised15-rays were produced by including a µ m -thick diamond phase plate upstream of the sam-ple position . The degree of circular polarisation achieved was greater than , and with antransmission of approximately .The tomographic projections were aligned with high precision as described in Ref. . Magnetic tomography
When a single circular polarisation projection is measured, the componentof the magnetisation parallel to the X-ray beam is probed, along with the electronic structure of thesample. To probe all three components of the magnetisation, projections were measured around arotation axis for two orientations of the sample . Generally, the magnetic contrast of a projection isisolated from other contrast mechanisms by measuring the same projection using circular left andright polarised light, where the sign of the magnetic contrast is reversed, and taking the differencebetween the two images. Here, a single X-ray polarisation is used for all measurements and, inorder to isolate the magnetic structure, projections with circularly left polarisation are measured at θ and θ + 180 ◦ . Between these two angles, the magnetic contrast is reversed, which can be usedto differentiate the magnetic contrast from the electronic contrast. Therefore, for the magnetic to-mography measurements, circular left polarisation projections were measured through ◦ aboutthe rotation axis, instead of through ◦ , as in standard tomography.The magnetisation (which is a three-dimensional vector field) was reconstructed using a two-step gradient-based iterative reconstruction algorithm, described in Ref. . The spatial resolutionfor each component of the magnetisation was estimated using Fourier Shell Correlation , and athree-dimensional Hanning low-pass filter was used to remove high-frequency noise. The spatial16esolution of the reconstructed magnetisation was found to be
97 nm ,
125 nm and
127 nm in the x − z , x − y and y − z planes, respectively .The magnetic vorticity was calculated according to Equation 1. The magnetisation wasnormalised to obtain the unit length, which was used to calculate the magnetic vorticity. Thethree-dimensional visualisations of the magnetic vorticity and magnetisation were performed withParaview.To consider the topology of the magnetisation in three dimensions, pre-images correspondingto different directions are plotted within the pillar. The difference between the magnetisation vectorand the m x = 1 direction is calculated using: δ px = (cid:18) m x | m | − (cid:19) + (cid:18) m y | m | (cid:19) + (cid:18) m z | m | (cid:19) (2)To plot the m x = 1 pre-image, we plot an isosurface for δ px = 0 . . This results in a tube ratherthan a line, which is necessary due to the finite spatial resolution and signal-to-noise ratio of themeasurement. Analytical models
To qualitatively interpret and understand the observed structures, we build aseries of 2+1 dimensional models, which allow comparing the observed magnetization structures,preimages and the vorticity with the ones derived from modeled vortex loops with different mag-netization structures. These models are similar to those used for description of hopfions in Ref. 51.They are based on the subdivision of the magnetic material volume into thin slices, lying in the x − y plane of a Cartesian coordinate system. The magnetisation in each slice can then be described17y a complex function w of a complex variable u = x + ıy by means of stereographic projection { m x + ım y , m z } = { w, − ww } / (1 + ww ) , where the over-line denotes complex conjugation, sothat u = x − ıy , ı = √− . Without loss of generality, any three-dimensional magnetisation distri-bution m ( x, y, z ) can be described by a function w = w ( u, u, z ) , which depends on the complexcoordinate u within each slice and the extra-dimensional variable z , identifying the slice.For realistic models, including at least the exchange and the magnetostatic interactions, noexact solutions for non-uniform w ( u, u, z ) are known. However, if the magnetostatic interactionis neglected and w ( u, u, z ) is assumed to be weakly dependent on z , two large families of exactsolutions exist for w ( u, u, z ) at a fixed z . These are solitons , which are meromorphic func-tions w ( u, u, z ) = f ( u, z ) , and singular merons , which are functions with | w ( u, u, z ) | = 1 or w ( u, u, z ) = f ( u, z ) / | f ( u, z ) | . Zeros of f ( u, z ) correspond to the centers of magnetic vortices (orhedgehog-like structures, if the magnetisation vectors are rotated by π/ in the x - y plane). Thepoles correspond to the centers of the magnetic antivortices (or saddles). From the stereographicprojection it follows that for solitons m z = 1 in the centers of the vortices and m z = − in thecenters of antivortices.An example of meromorphic functions are the rational functions of a complex argument(quotient of two polynomials). They allow direct expression of the vortex/antivortex pair annihila-tion as a cancellation of two identical monomials, whereas creation is a time-reversed process. Thenumber of vortices in each slice is a conserved quantity (topological charge, or skyrmion number)in the sense that it cannot be changed by a smooth singularity-free variation of the magnetisation18istribution. For the slices in the x - y plane the topological charge density is the z -componentof the vorticity Ω z and the total charge is the integral of this density over the whole slice. Cre-ation and annihilation of the vortex-antivortex pairs within the soliton is always accompanied by asingularity.A vortex ring can be understood as a process of creation, separation, convergence and an-nihilation of a vortex-antivortex pair as the variable z advances through the successive slices .Consider w BPr ( u, u, z ) = f ( u, z ) = ı u − p ( z ) u + p ( z ) = ı u − (cid:112) − ( z/ u + (cid:112) − ( z/ (3)for an (arbitrary) range − < z < , where the specific expression for p ( z ) was chosen to make thevortex and antivortex cores extend along arcs, as in the experimental data. It describes the creationof a vortex-antivortex pair at x = y = 0 and z = 2 , the vortex and antivortex moving apart (with themaximum distance between their centres equal to at z = 0 ), then approaching each other again,and annihilating at z = − . We call this model the Belavin-Polyakov ring because each slice is aBelavin-Polyakov soliton, described by a meromorphic w ( u, u, z ) . The corresponding schematicmagnetisation, set of preimages and vorticity are shown in figure M1a. A similar preimage patternsconnecting two Bloch points were indeed observed in our sample. However, the correspondingvorticity distributions are different. Indeed, instead of a single centrally-symmetric vorticity bundlewe reconstruct a pair of bundles, corresponding to the vortex and antivortex centers. Clearly, thepure Belavin-Polyakov ring model can not reproduce this feature.19igure M1: Top to bottom: Magnetisation, pre-images and vorticity distribution for the different dimensional analytical models, discussed in the methods section. The magnetisation plotsonly include the projection of the magnetisation onto the shown planes, while the rings correspondto the positions of the vortex and antivortex centers, and the color indicates the m Z component ofthe magnetisation. The preimages are shown as volumes where the magnetisation vectors deviateonly slightly from certain directions d i , indicated by the color-coded arrows on each correspondinglegend. The opacity and color on the vorticity plots indicates the magnitude of local vorticityvectors. The structure in c is comparable to the vortex rings in figure 2, while the structure in d iscomparable to that in figure 3. 20o ’unbundle’ the vortex and antivortex, we can use the instanton model by writing: w i ( u, u, z ) = f ( u, z ) /c ( z ) | f ( u, z ) | ≤ c ( z ) f ( u, z ) / | f ( u, z ) | d ( z ) > | f ( u, z ) | > c ( z ) f ( u, z ) /d ( z ) | f ( u, z ) | > d ( z ) , (4)where d ( z ) = 1 /c ( z ) , assuming the same size for the vortex and antivortex cores. Choosing c ( z ) = 1 − q + q | z | / < allows the control of the size of the vortex and antivortex cores(where m z (cid:54) = 0 ) at the central plane z = 0 via the parameter q . The magnetisation, preimages andvorticity for such an instanton ring with q = 3 / are shown in figure M1b. While they reproducequalitatively both the vorticity distribution and the preimages, shown in figures 3b and 3g, thestructure of the Bloch points is different. Indeed, the instanton ring has two hedgehog-type Blochpoints (in which the magnetisation directions are opposite), whereas the observed structure, shownin figure 3, contains two different types of Bloch points. Additionally, this model differs fromthe observation in figure 3 in that singularities are absent at the transition from the experimentally-observed vortex and antivortex pair to a uniformly-magnetized region. The Bloch points in figure 3rather coincide with the polarisation reversal of vortex and antivortex cores as they propagatethrough the volume of the sample. In order to analytically describe this structure, we first need tobuild a model for a vortex ring.To describe a vortex-antivortex pair unbound by Bloch point singularities, the vortex and theantivortex must have identical polarisations (i.e. the same direction of m z within the core). Inthis case the topological charge in each slice is zero. Such a configuration can be obtained as a21eneralisation of (4) w r ( u, u, z ) = A ( z ) f ( u, z ) /c ( z ) | f ( u, z ) | ≤ c ( z ) f ( u, z ) / | f ( u, z ) | d ( z ) > | f ( u, z ) | > c ( z ) d ( z ) /f ( u, z ) | f ( u, z ) | > d ( z ) , (5)where the modification to the last line reverses the polarisation of the antivortex. The factor A ( z ) =(1 − z / s ensures that, at z = ± , the function w r = 0 , which corresponds to the uniform state.The parameter s allows for the control of the degree of quasiuniformity: the smaller s is, the less m z deviates from . The magnetisation, preimages and vorticity for such a quasiuniform ringwith q = 3 / and s = 1 / are shown in figure M1c. They are qualitatively analogous to theexperimentally-observed vortex rings in figures 2b and 2d.Finally, we can extend the above model to a vortex ring in which the polarisation reversesalong the vortex and the antivortex cores, in the presence of Bloch points. To describe this state,we note that with s = 1 , c ( z ) = z / , the magnetisation of the quasiuniform ring (5) at z = 0 lies completely in the x - y plane except for at the centres of the the vortex and antivortex, where itsdirection is undefined. Joining at the central plane two half-rings with opposite polarisations: w vls ( u, u, z ) = A ( z ) w r ( u, u, z ) z ≤ /w r ( u, u, z ) z > (6)yields the model for the vortex loop with Bloch point singularities, shown in figure M1d. Thestructure corresponds well to the observations in figure 3, including the observed Bloch pointtypes. 22igure M2: Detailed overview of the vortex ring, shown in successive slices through the loop.The colourscale in the top row indicates the magnetisation, while the colourscale in the bottomrow indicates the vorticity. The vorticity associated with the vortex structure extending throughoutthe pillar changes in sign in slice d due to the presence of a Bloch point, while the vortex-antivortexpair conserves its vorticity throughout. In slices b and c , the magnetisation forms a cross-tie walllike structure, which dissolves as the pair unwinds, at slices a and d , leaving the a single vortex.Note that despite piecewise nature of the above functions, the resulting magnetisation vectorfields are continuous (apart at the Bloch points). While neither ansatz in the presented series is anexact solution of the corresponding micromagnetic problem (not even of its restricted exchange-only version), they provide a simple and easily interpretable model to understand the observedmagnetisation distributions. 23igure M3: Detailed overview of the magnetic state of the vortex loop containing Bloch points,shown in successive slices through the loop. The colourscale in the top row indicates the mag-netisation, while the colourscale in the bottom row indicates the vorticity. The vorticity along thevortex core reverses between slices b and c , while the vorticity along the antivortex core reversesbetween slices c and d . 24igure M4: Plotting regions of significant magnetic vorticity reveals the effect of different fieldhistories on the network of vortex-antivortex structures present in a GdCo micropillar a) followingthe application of a 7 T saturating field and c) following saturation and field cooling. Regions ofhigh divergence of the magnetic vorticity indicate the presence of Bloch points (red) and anti-Blochpoints (blue) (b) at remanence, following saturation and d) after heating at 400 K and field coolingin a 7 T field. Noticeably fewer magnetic structures with high vorticity are present after the fieldcooling procedure in than after the simple application of a magnetic field.25 Contributions
The study of topological magnetic features in three dimensions was conceived by S.G., C.D. andK.L.M., and originated from a larger project on three-dimensional magnetic systems conceivedby L.J.H and J.R.. C.D., M.G.-S., S.G., V.S., M.H. and J.R. performed the experiments. Magne-tometry measurements of the material were performed by N.S.B.. C.D. performed the magneticreconstruction with support from M.G.-S. and V.S.. C.D. analysed the data and N.C. conceivedthe calculation of the magnetic vorticity. C.D., S.G., K.L.M. and N.C. interpreted the magneticconfiguration. K.L.M. developed the analytical model. C.D., K.L.M., N.C. and S.G. wrote themanuscript with contributions from all authors.
X-ray measurements were performed at the cSAXS beamline at the Swiss Light Source, PaulScherrer Institut, Switzerland. The authors are grateful to Andrei Bogatyr¨ev for his careful readingof the manuscript and many valuable remarks. We thank R. M. Galera for providing and perform-ing magnetic characterisations of the GdCo nugget, S. Stutz for the sample fabrication, and E.M¨uller from the Electron Microscopy Facility at PSI for the FIB-preparation of the pillar sam-ples. C.D. is supported by the Leverhulme Trust (ECF-2018-016), the Isaac Newton Trust (18-08)and the LOral-UNESCO UK and Ireland Fellowship For Women In Science. S.G. was fundedby the Swiss National Science Foundation, Spark Project Number 190736. K.L.M. acknowledgesthe support of the Russian Science Foundation under the project RSF 16-11-10349. N.C. was26upported by EPSRC Grant EP/P034616/1 and by a Simons Investigator Award. The authors declare no competing financial interests.
Correspondence to C.D., K.L.M. or S.G.
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